1.我不介意

I don't mind.

It is often used to answer "Do you mind"

2.我不管 and 我不在乎 can be translated as "I don't care"

But there are some gentle difference.

And I think Cheshire's example is great.

is an irresponsible way,which express your attitude toward something.

3.最后and结果

最后 can be an adverb, means at last

结果 can be an adverb,too.but hardly do we use this way.It you see it,it often mean a little surprise result.

ex:最后,他终于承认了. At last/eventually/at the end,he admitted.

结果,他(竟然/居然)放弃了. At last,he (should) gave up.

BUT you should know that 结果 is a noun,too.it means result.

I think you will meet the expression"后果",it's similar to "结果",and it often express something bad.and in enlish,we use consequence.

ex:事情的结果是她原谅了他. The result is she forgave him.

这将造成灾难性的后果. this will cause catastorpic consequence.

"我不介意" - I don't mind --(A)

"我不管" - I don't care -- (B)

The meaning of "我不在乎" is somewhat in between (A) and (B)

(A) sounds nice 

(B) can sound rude and strong depending on how you use it. 

"我不在乎" is a bit netural, but it can still lean toward (A) or (B) depending on how you use it. 

If you want to conclude a series of events, "最後" and "結果"  can be used interchangeably in the following way:

Ex. 

我昨晚下班晚了 (first event), 又沒能趕上最後一班車 (second event), 最後我到家的時候已經是凌晨兩點了 (last event). 

Ex.
我昨晚下班晚了 (first event), 又沒能趕上最後一班車 (second event), 結果我到家的時候已經是凌晨兩點了 (last event).

In the following sentences,  "最後" and "結果" are not interchangeable:

王先生住在這棟樓的最後面. 

會議都已經開始半小時了, 老王最後才到. 

我們兩個人的愛情這樣下去是不會有結果的. 

老王的農作物都已經開花結果了, 今年會是一個豐收的一年. 

1)

我不介意 = "I don't mind";

you say it when someone thinks he or she might cause you inconvenience

e.g. 我不介意 you borrow a pencil from me, I've got plenty.

我不管 is more or less 我不在乎 = "I don't care";

it can be a strong tone so only say it to show your strong will

e.g. 我不在乎 whether you will return that pencil to me, I don't like that black one. I don't like you either.

2)

最後 and 結果 are interchangeable, not much difference really.

If it has to be precise, 最後 speaks sequence-wise as 最後 by literal means "the last" while 結果 focuses on the very outcome as 結果 by literal means "to bear fruit (in botany)".

e.g. "We got lost the first four hours in the mountains, so stupid!" "It wasn't even the worst part!" "結果呢？" Mum asked. Not listening, Tom went on, "Then the rain poured down on us all of a sudden, what the heck! Strangely enough, a troop of wild monkeys out of nowhere yelled at us and one of them as I remember standing high, went pee-pee on Zoe's head as soon as the rain stopped!" Mum glanced at the wall clock once again, indifferently as well as impatiently, "結果呢？" Tom, enthusiastically, "And then the 5th sun exploded again, right on our way to Dark Valley. Holy shoot! Somebody should have fixed it! Before we came to ourselves, we found Schneider naked, all his shirt, pants, shoes, everthing, flew away and scattered because of the sound waves!" Mum started to look daggers at Tom, "結、果、呢？" Tom slowly stumbled to what mum meant, "Oh 結果？結果啊！We 最後 managed to tear the hahajumba down. A bunch of it, man! Anyways we're back safe and sound. Now it is all processed and in the refrig waiting to serve you." Mum headed to the refrigerator on the 123456789th floor of her mansion in Mare Moscoviense downtown almost right before Tom finished his last word, "Very good Tom, that's my son! I'm going hella haha now," still in speed of light, "Yoo-hoo, hahajumba! I'm coming!" A voice fading away far far away, "Yes, mum. Bon appe..."

謝謝大家! Thank you for the very detailed explanation! Now with all these examples I have a much clearer idea of the different uses between these words.  Now I should put them into practise!